Periodic structures

Recently, new material processing techniques have been used to produce solids with periodic structures to create meta-materials for improved or desired functionality. For instance, Piccione & Gianola (2015) fabricated nanomeshes, a periodic array of squares with a circular hole inside exhibit higher thermoelectric responses than that of crystal silicon. An example of such a structure is sketched in Fig.5.12.

Figure 5.12: Periodic structure with a repeating unit cell ABCD.

In order to determine the macroscopic electromechanical response of this periodic struc- ture, we study the behavior of the unit cell ABCD using appropriate periodic boundary conditions as described in the following.

Let ¯F be the macroscopic deformation field in the periodic structure. The presence of the hole in the unit cell perturbs the displacement filed locally in the unit cell. In fact, the displacement fieldu(x) in the unit cell can be written in the form

ui(x)=(F¯ij−δij)xj+u∗i(x), (5.63)

whereu∗(x) is a periodic function with zero mean deformation gradient on the unit cell. We denote the quantities on sidesAB,BC,CD and DAwith superscriptl,b,r andu, respectively. Then periodicity requires

u∗u =u∗b, u∗l=u∗r. (5.64)

Hence, the total displacement field satisfies the conditions

uu_i −ub_i =(F¯ij−δij) (xuj −xbj) and uri −uli =(F¯ij−δij) (xrj−xlj). (5.65)

LetL be the length of the sides of the square unit cell. Then

xu_j −xb_j =δj2L, xrj−xlj =δj1L, (5.66)

so that

uu_i −ub_i =(F¯i2−δi2)L and uri −uli=(F¯i1−δi1)L. (5.67)

Similarly, for the electric field, we have

φu,r−φb,l=E¯j(xu,rj −x b,l

j ), (5.68)

where ¯E is the macroscopic electric field. The macroscopic fields ¯F and ¯E are the fields that develop in the structure when there are no microscopic holes.

We use the finite element method to study the response of the square unit cell when subject to mechanical and electrical loads. Since the displacement gradients αij = ui,j are treated as independent degrees of freedom in the finite element formulation, similar

periodicity conditions are imposed onα in the numerical solution:

αu−αb=αr−αl=F¯−δ. (5.69)

The periodicity conditions are imposed in ABAQUS through a “user MPC” subroutine. In the following we present results for the case in which the macroscopic loads on the unit cell are a normal strain ¯ε22in thex2−direction and electric field ¯E2, also in thex2−direction,

which is created by opposite charges±ω¯ at on the top and bottom surfaces of the unit cell. Calculations are carried out for the following parameters

{ν, a ǫ0, ℓ L, ℓ R}={0.30, 0.0018, 1 12, 1 3} (5.70)

and various values of ˜˜f1 = f˜1/(R√a µ) and ˜˜f2 = f˜2/(R√a µ). Here R is the radius of the

hole andLthe length of the sides of the unit cell. This creates a meta-material with defect volume fraction of π/16≈19.6%. 1 1.2 1.4 1.6 1.8 2 0.8 1 1.2 1.4 1.6 1.8 x1/R ε22 / ¯ ε22 Flexoelectric, ˜f1= 0.10,f˜2= 0.05 Flexoelectric, ˜f1= ˜f2= 0.05 Electrostatics (a) 1 1.2 1.4 1.6 1.8 2 1.2 1.4 1.6 1.8 2 2.2 x1/R P2 / [( ǫ − ǫ0 ) ¯E2 ] (b)

Figure 5.13: Variation of the opening normal strain ε22 (a) and polarization P2 (b)

along the x1−axis ahead of the void due to a macroscopic strain ¯ε22 and an electric

field ¯E2 in the x2−direction, unit-cell under tension

Figure 5.13 shows the variation of the normal strain ε22 and the polarization P2 along

the x1−axis ahead of the hole. Figure 5.13 shows also the corresponding solutions of SGE

are varied.

We consider next the problem in which the macroscopic loads on the unit cell are a shear strain ¯ε12 in the x2−direction and electric field ¯E2 created by opposite charges ±ω¯

at on the top and bottom surfaces of the unit cell. Now the problem is expected to be anti-symmetric about the x1−axis, and due to the flexoelectric coupling, some interesting

effects are observed.

1 1.2 1.4 1.6 1.8 2 1.05 1.1 1.15 1.2 1.25 1.3 x2/R ε12 / ¯ ε12 Flexoelectric, ˜f1= 0.10,f˜2= 0.05 Flexoelectric, ˜f1= ˜f2= 0.05 Electrostatics (a) 1 1.2 1.4 1.6 1.8 2 −5 0 5 10 15 20x 10 −3 x2/R P1 / [( ǫ − ǫ0 ) ¯E2 ] (b)

Figure 5.14: Variation of shear strain ε12 (a) and polarization P1 (b) along the x2−

axis above of the void due to a macroscopic strain ¯ε12 and an electric field ¯E2 in the

x2−direction, unit-cell under shear.

Figure 5.14 shows the variation of the shear strain ε12 and the polarization P1 along

the x2−axis above the hole. Figure 5.14 shows also the corresponding solutions of SGE

and electrostatics. The strain profile of ε12 is affected by flexoelectricity and the relative

effect is related to the magnitude of the flexoelectric constant. The effect is highly localized around the hole or meta defect whereas the overall profile ofε12 is relatively flat and small

gradient is developed, at least along this axis. More interestingly, due to the coupling of strain gradient and polarization, extra polarization alongx1 direction is produced. In other

words, flexoelectricty rotates the polarization field towards x1−axis. Other polarization

rotation phenomena can also be found in the works of Catalan et al. (2011), Lu et al. (2012); they are realized in ferroelectric thin films and are believed to have applicability in memory devices. Here, however, we have demonstrated flexoelectric rotation of polarization in periodic meta-structures.

The analyses of the periodic meta-structure and the elliptical hole problems suggest an alternative way of studying flexoelectricity. Recall that due to Timoshenko & Goodier (1969), the classical solution of a circular hole in a infinite elastic body (under uniaxial tension) predicts a stress concentration factor of 3 and that strain/stress decays to the far field level ¯ε as (r/R)−2. Therefore, a good estimate of strain gradient around the hole, wherer∼R, can be calculated through

∣κ˜∣≃ηε¯

R (5.71)

where η is the concentration factor. Therefore, these periodic structures can generate considerably large strain gradient, when the holes are small. Reducing the size of the hole produces greater strain gradient without increased deformation, which sometimes causes inelastic behavior of the material. Indeed, periodic nano-scale or even atomistic scale holes have already been observed to alter the electromechanical behaviors of certain 2D materials Zelisko et al. (2014). However, holes of these scales are difficult to make; experimental observations are possible only due to certain inherent atomic structures. On the other hand, for meta-materials, the size of the hole can be in the range of hundreds of nanometers as in Piccione & Gianola (2015), which, by the above analysis, can also produce large gradients. For these structures, we can design the arrangement, size and spacing so as to meet different needs. Combining precise mechanical and electrical probes, it is possible to demonstrate how flexoelectricity can be used to alter material properties in larger and realizable scales as well.

Moreoever, this periodic structure can be a new source where we can measure flexoelec- tric constants. So far, the most reliable means to measure them is through beam bending experiments. These, however, cannot determine all components of the flexoelectric tensor (even the simplest isotropic one) as pointed out by Zubko et al. (2007). Therefore, alter- native measurement techniques are required. The truncated pyramid structure is one such technique, but due to non-trivial deformation concentration around the edges, this method can hardly measure the correct flexoelectric constant. Besides, recent study of Hong & Vanderbilt (2013) predicted that some materials (such as, silicon) have finite volumetric flexoelectric constant ˜f, but vanishing or very small bending flexoelectric constants. The

beam bending experiments are of little use for such materials, but the periodic structure studied here could be an ideal set up to overcome these difficultes. It gives a large gradient, but within a smooth profile (without singular fields due to sharp edges). The magnitude of the gradients can be easily controlled by altering loading or geometry (without exceed- ing the elastic limit). Both ˜f and ˜f1 (isotropic case) appear in the solutions, so we can

determine them by appropriate measurements.

5.7

Concluding remarks

In this chapter, we have formulated a variational form that is completely consistent with the continuum theory of flexoelectricity. The form utilizes a mixed formulation and circumvents the difficulties of modeling gradient effects in flexoelectric solids by introducing extra degrees of freedom. This variational form is general and can incorporate the piezoelectric effect as well. A new element is developed for adapting the variational form to finite element calculation. The known analytic solution of a pressurized tube is employed as a benchmark problem for validation. Then the method is used to study three types of problems which are beyond current analytic capability. Asymptotic theories of cracks are confirmed and a more precise description of the fracture landscape is accomplished. Single hole in an infinite medium as well as periodic meta-structures illustrate the non-trivial coupling of electric loading and deformation. They also offer further inspiration for alternative means of measuring and utilizing flexoelectricity.

Chapter 6

Closure

This dissertation investigated continuum and computational modeling techniques for flex- oelectric solids. Governing equations were derived by a generalized Toupin’s variational principle and then used for analyzing interesting boundary value problems. They were also employed to study defects, where strong gradients are known to exist. A “mixed” finite element scheme was formulated based on the continuum theory. This computational tool gave insights into problems in more complicated geometries.

We began the investigation by a review of classical continuum theories on electrome- chanics. The governing differential equations of the classical theory follow as a result of Toupin’s variational principle. By adding strain gradients, this principle was generalized to study flexoelectricity. It was recognized that higher-order stresses give extra contributions to the physical stresses, in the bulk, as well as, on the boundary. A reciprocal theorem was proved for linear flexoelectricity. We then restricted ourselves to an isotropic flexoelectric material and derived Navier type governing equations. The flexoelectric effect raised the order of the governing differential equations and altered the effective length scales in the equations. Based on the formulation, it was realized that gradient elasticity is an essential part of flexoelectricity, and it gives upper bounds on the flexoelectric coupling constants.

Using the above theory, we then looked into four different boundary value problems that are closely related to experiments. We first studied the problem of beam bending, which is one of the most common ways of utilizing flexoelectricity. We pointed out that the measurements using this approach gave the bending flexoelectric constant, a linear

combination of transverse and longitudinal flexoelectric constant. We showed that applying an electric voltage across the beam thickness resulted in bending moments. In addition, a size-dependent flexoelectric stiffening between short circuited and open circuited beam followed from our analysis. We then considered a circular shaft under torsion and found that it does not create any flexoelectric coupling effect, if it is made of isotropic or cubic material. Our analysis of a pressurized cylinder shed light on how to control stress concentration in flexoelectric materials. We found that flexoelectricity causes an azimuthal polarization in cylinders under shear.

The theory was used to examine the interplay of flexoelectricity with defects. This interplay is intriguing because defects create strong gradients, and hence magnify the flex- oelectric coupling effect. Our analysis revealed that the presence of a non-charged point defect in a flexoelectric solids creates a Yukawa type of electric potential field around it through flexoelectric coupling. This electric potential decays exponentially in the far field and can only be observed when defect size is in the same range as the flexoelectric length scale. When it comes to dislocations, we learned that screw dislocations in isotropic or cubic flexoelectric solids do not polarize (in agreement with experiments), but edge dislocations do. We calculated the polarized charges and electric field due to an edge dislocation. Our estimates agreed with experiments in alkali salts and ice.

For cracks, we showed that the leading order of the asymptotic field in flexoelectric solids is altered due to gradient effects. Also, gradient effects result in additional intensity factors to characterize the full fracture behavior of the solids. We found that the crack opening profile changes to a cusp-like shape, rather than the parabolic shape in linear fracture mechanics. A path independent J integral can still be calculated, but in a slightly different manner. These integrals show that, similar to piezoelectric solids, flexoelectric coupling reduces the energy release rate so that more energy must be supplied in order for a crack to grow.

To deal with more complicated geometries, we designed a finite element scheme that is consistent with the continuum theory. The scheme utilizes a “mixed” formulation and treats displacement and displacement gradient as separate variables, but their relation is enforced in an integral sense. A completely equivalent Type I gradient elasticity is used in the formulation. Effects of higher-order stresses are introduced by some virtual stresses

conjugate to strain gradients. Based on Toupin’s variational principle, we derive a weak form for this “mixed” formulation. A special 9-node element is designed to implement the formulation. The finite element code is then written for isotropic flexoelectric solids and validated through patch test and benchmark problems.

This computational tool helps us dig into three types of complicated structures. First, we studied an elliptical hole in a plate. In that problem, we observed that net flexoelectric polarization can be induced by mixing electrical and mechanical loads. This could help us understand the flexoelectric polarization associated with the evolution of micro-voids in solids. Then we studied an edge crack panel and compared our finite element result with that of the asymptotic analysis described above. We found that the singularity predicted by the asymptotic theory is correct, but the theory is valid only very close to the crack tip. Our finite element analysis also helped visualize the changes in symmetry properties of fields around the crack tip due to mixing modes–another prediction from the asymptotic theory. Last, a structure with periodic holes in it was studied. Periodic boundary conditions were imposed to compute an average response of this solid. Materials with large flexoelectric coupling constants can generate large polarization in these structures without perturbing the deformation significantly. We showed that flexoelectric rotation is also possible by applying shear to the structure. This meta-defect structure has potential as an alternative for measuring and utilizing flexoelectricity.

In spite of all this work the theoretical development of this subject is far from complete. Our work relied on the small deformation assumption, but finite deformation kinematics will generalize the applicability of the theory to soft materials. Another aspect that is not discussed in this work is dynamics. How does flexoelectricity affect wave propagaton? Can we examine wave propagation in structures with periodic cells? Can we look at dynamic crack growth in these solids? An intriguing possibility is that dynamically moving cracks in flexoelectric solids can give rise to electro-magnetic radiation due to time varying electric fields. Can we study such phenomena? Also of interest are problems in which flexoelectricity affects the stability of structures, e.g., buckling of ribbons and plates.

In conclusion, we have studied flexoelectricity starting from fundamental variational princples and derived consistent governing equations and proper boundary conditions. We have solved important problems analytically to compare with experiments and motivate

application. We have studied defects and their interplay with flexoelectricity in this fashion. Our studies on cracks give insights into the fracture behavior of flexoelectric solids. We have developed a finite element method that is consistent with the flexoelectric continuum theory using a “mixed” formulation. We have applied it to certain problems of interest and predicted the effective behavior of complicated structures.

Appendix A

Appendix of Chapter 2

A.1

Example of reciprocity

We will use our solution to the flexoelectric beam to demonstrate the reciprocal theorem. Consider two problems (see figure A.1) for two clamped-clamped beams with exactly same geometries (thickness 2h, width w and length L). In the original problem, a force Q is exerted atx1=L1; in the reciprocal problem, a constant voltage V is prescribed across the

beam fromL2 to L. We model this voltage as a step function and neglect edge effects. The

deflection profile of the beam is u2(x1). The variables in the original problem will have

upper index 1 and those in the reciprocal problem will have upper index 2.

We will start with the reciprocal problem. Since there is no distributed load along the beam, Eqn(2.59) and (2.58) give:

κ(2)=d 2_u(2) 2 dx2 1 = fˆbA aGE E2H(x1−L2)= E2 Vb H(x1−L2) (A.1) Q L L₁ V=0 (a) V L L₂ (b)

Figure A.1: In (a), a point loadQis applied, while in (b) there is a potential difference

where Vb =aGE/(fˆbA) is introduced to avoid redundant repetition of constants and H(.) is the unit step function. We will use the Macaulay bracket ⟨.⟩n to denote the nth anti- derivative of H(.). By applying the clamped boundary conditions at the two ends, the deflection profile in the reciprocal problem can be calculated:

u(₂2)(x1)=− E2 Vb [− 1 2⟨x1−L2⟩ 2_+L 2(L−L2) x3₁ L3 + (L−L2)(L−3L2) x2₁ 2L2] (A.2)

Using this we are able to determineW(12)_:

W(12)_=Qu(2) 2 (L1)= QV 4hVb [−⟨ L1−L2⟩2+2L2(L−L2) L3₁ L3 + (L−L2)(L−3L2) L2₁ L2] (A.3)

For the original problem the deflection and electric displacement are given by:

u(₂1)(x1)= Q 6GE [−( L−L1)2(L+2L1) x3₁ L3 +3(L−L1) 2_L 1 x2₁ L2 + ⟨x1−L1⟩ 3_{], (A.4)} D(₂1)(x1)= ˆ fb aκ (1)₌ Q AVb [−( L−L1)2(L+2L1) x1 L3 + ( L−L1)2L1 L2 + ⟨x1−L1⟩] (A.5)

Hence, we are able to calculate W(21) _as:

W(21)_=−w ∫_LL₂ D(₂1)V dx1= QV 4hVb[ L2(L−L1)2 L3 (2L1L−2L2L1−LL2) + ⟨L2−L1⟩ 2_{] (A.6)} Note that⟨L1−L2⟩2+ ⟨L2−L1⟩2=(L1−L2)2, soW(12)= W(21).